How can I delete the same animation/curve channel from all selected objects?
Find and remove
The fcurves collection of an action has a method to find fcurves from data path and index, and a remove method.
The same action can be linked to multiple objects, so will use a set to find all the actions' fcurves.
There really is no need to look at the group, location x keyframes have data path 'location' and array_index 0.
Test code. Since you mention you are going to convert this to an operator have added the context variable context = bpy.context  Once test code works, lose this and use the context argument passed to operator/panel methods, when pasting into operator code.
import bpy

context = bpy.context

data_path = &quot;location&quot;
index = 0 # location.x or location[0]

#objects = bpy.data.objects # all
#objects = context.scene.objects # all in scene
objects = context.selected_objects # selected

all_fcurves = set(
    o.animation_data.action.fcurves for o in objects
    if o.animation_data and o.animation_data.action
    )

for fcurves in all_fcurves:
    fcurve = fcurves.find(data_path, index=index)
    if fcurve:
        fcurves.remove(fcurve)

for bones the data path is akin to
'pose.bones[&quot;Bone&quot;].location`

in which case to find  x location keyframe it could be simpler to replace the find method with
def find(fcurves, data_path, index):
    x = [fc for fc in fcurves 
         if fc.data_path.endswith(datapath) 
         and fc.array_index == index]
    return x.pop() if x else None


Thanks, I can use your answer later, but you missed the exact point of
my question. I do not know that the user will delete specifically the
X Location channel. Instead, I want to present a list of channels from
all selected objects to the user, so that he will select, for example,
&quot;I want to delete the Z Location from all selected objects&quot;. However,
I could not find a way to convert location[2] to &quot;Z Location&quot;.

Re-read the question appears to be a pretty good attempt at an answer to part i. Generally the idea is to have one question per question , re part 2
Can get the name of the property via the &quot;RNA&quot; eg if an object has an fcurve with datapath &quot;location&quot; or &quot;rotation_euler&quot;
&gt;&gt;&gt; C.object.bl_rna.properties['location'].name
'Location'

&gt;&gt;&gt; C.object.bl_rna.properties['rotation_euler'].name
'Euler Rotation'

&gt;&gt;&gt; C.object.bl_rna.properties['rotation_euler'].subtype
'EULER'

&gt;&gt;&gt; C.object.bl_rna.properties['location'].subtype
'TRANSLATION'

As for the axis name from index, have asked this question Python RNA getting the names of Vector axes. (eg &quot;g&quot; is color[1])
&gt;&gt;&gt; index = 0
&gt;&gt;&gt; &quot;XYZ&quot;[index]
'X'

and as for your third question, with espanol selected as language.
&gt;&gt;&gt; bpy.app.translations.pgettext(&quot;Location&quot;)
'Posición'

consult the docs.  https://docs.blender.org/api/current/bpy.app.translations.html#module-bpy.app.translations